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Question
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab.
Solution
Let m1 and m2 be the slopes of the lines given by ax2 + 2hxy + by2 = 0
∴ m1 + m2 = `- "2h"/"b"`
and m1m2 = `"a"/"b"`
We are given that m2 = 4m1
∴ m1 + 4m1 = `- "2h"/"b"`
∴ 5m1 = `(- "2h")/"b"`
∴ m1 = `- "2h"/"5b"` .....(1)
Also, m1(4m1) = `"a"/"b"`
∴ `4"m"_1^2 = "a"/"b"`
∴ `"m"_1^2 = "a"/"4b"`
∴ `("- 2h"/"5b")^2 = "a"/"4b"` ...[By(1)]
∴ `"4h"^2/"25b"^2 = "a"/"4b"`
∴ `"4h"^2/"25b" = "a"/4`, as b ≠ 0
∴ 16h2 = 25ab
This is the required condition.
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